| L(s) = 1 | + 1.07·3-s − 1.81·5-s − 7-s − 1.84·9-s + 4.90·11-s − 4.83·13-s − 1.94·15-s + 17-s + 5.27·19-s − 1.07·21-s + 3.94·23-s − 1.71·25-s − 5.20·27-s − 2.75·29-s + 1.28·31-s + 5.27·33-s + 1.81·35-s + 0.798·37-s − 5.18·39-s + 4.50·41-s − 5.71·43-s + 3.34·45-s − 3.76·47-s + 49-s + 1.07·51-s − 7.86·53-s − 8.90·55-s + ⋯ |
| L(s) = 1 | + 0.619·3-s − 0.810·5-s − 0.377·7-s − 0.615·9-s + 1.48·11-s − 1.34·13-s − 0.502·15-s + 0.242·17-s + 1.20·19-s − 0.234·21-s + 0.822·23-s − 0.342·25-s − 1.00·27-s − 0.512·29-s + 0.230·31-s + 0.917·33-s + 0.306·35-s + 0.131·37-s − 0.830·39-s + 0.704·41-s − 0.871·43-s + 0.499·45-s − 0.549·47-s + 0.142·49-s + 0.150·51-s − 1.08·53-s − 1.20·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 1.07T + 3T^{2} \) |
| 5 | \( 1 + 1.81T + 5T^{2} \) |
| 11 | \( 1 - 4.90T + 11T^{2} \) |
| 13 | \( 1 + 4.83T + 13T^{2} \) |
| 19 | \( 1 - 5.27T + 19T^{2} \) |
| 23 | \( 1 - 3.94T + 23T^{2} \) |
| 29 | \( 1 + 2.75T + 29T^{2} \) |
| 31 | \( 1 - 1.28T + 31T^{2} \) |
| 37 | \( 1 - 0.798T + 37T^{2} \) |
| 41 | \( 1 - 4.50T + 41T^{2} \) |
| 43 | \( 1 + 5.71T + 43T^{2} \) |
| 47 | \( 1 + 3.76T + 47T^{2} \) |
| 53 | \( 1 + 7.86T + 53T^{2} \) |
| 59 | \( 1 + 0.722T + 59T^{2} \) |
| 61 | \( 1 + 8.29T + 61T^{2} \) |
| 67 | \( 1 + 0.598T + 67T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 + 7.53T + 73T^{2} \) |
| 79 | \( 1 + 0.156T + 79T^{2} \) |
| 83 | \( 1 - 6.38T + 83T^{2} \) |
| 89 | \( 1 + 0.304T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.005436328883148807493657774879, −7.52166405178376635301381609837, −6.83553897050101447754137725694, −5.93807492142009468922970216185, −5.01890694427427408822783295610, −4.13089930324050149293044638342, −3.34563087625857584241152384107, −2.77140603684487745880483767053, −1.44642416367983555411354163440, 0,
1.44642416367983555411354163440, 2.77140603684487745880483767053, 3.34563087625857584241152384107, 4.13089930324050149293044638342, 5.01890694427427408822783295610, 5.93807492142009468922970216185, 6.83553897050101447754137725694, 7.52166405178376635301381609837, 8.005436328883148807493657774879
